Download OU B.Sc 5th Year 2019 B.Sc Maths Question Bank Question Paper

Download OU (Osmania University) B.Sc Fifth Year (5th Year) B.Sc Maths Question Bank Question Paper

This post was last modified on 21 November 2019

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l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

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v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

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m
v=
{lS]=1
3

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0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch

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l?2
?--2andv;=l7

Determineif{vbv?isabasicforR?.ls{vbv?a
3?9

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basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.

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l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.

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...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II

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.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

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FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I

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I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=

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.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=
iii

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Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m
v=

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{lS]=1
3
0.51
z
o}

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a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

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Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.

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Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find

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.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.

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.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace

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ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.
2L

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22.
23.
24.
25.
.Letu=l?

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FindvinR?suchthatl
l?3~l
=T
2438]uv.

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.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6

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-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0

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Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i

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._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l

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LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let

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T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,

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T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

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FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I

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I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=

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.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=
iii

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Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m
v=

--- Content provided by‍ FirstRanker.com ---

{lS]=1
3
0.51
z
o}

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a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

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Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.

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Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find

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ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.
2L

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22.
23.
24.
25.
.Letu=l?

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FindvinR?suchthatl
l?3~l
=T
2438]uv.

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.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6

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-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0

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Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i

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._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l

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T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

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26.
27.
28.
29.

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30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).

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P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.

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b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li

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Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

Constructthegeneralsolutionofx?=Ax.A=i

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_?

1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

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FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I

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I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=

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.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=
iii

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Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m
v=

--- Content provided by FirstRanker.com ---

{lS]=1
3
0.51
z
o}

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a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

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Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.

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Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find

--- Content provided by‌ FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.
2L

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22.
23.
24.
25.
.Letu=l?

--- Content provided by‌ FirstRanker.com ---

FindvinR?suchthatl
l?3~l
=T
2438]uv.

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.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6

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-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0

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Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i

--- Content provided by FirstRanker.com ---

._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l

--- Content provided by⁠ FirstRanker.com ---

T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

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26.
27.
28.
29.

--- Content provided by FirstRanker.com ---

--- Content provided by‍ FirstRanker.com ---

P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.

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b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li

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Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

Constructthegeneralsolutionofx?=Ax.A=i

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_?

1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

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2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry

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Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:

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(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres

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1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres

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.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.

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whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H

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Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra

--- Content provided by⁠ FirstRanker.com ---

Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.

--- Content provided by⁠ FirstRanker.com ---

.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=

--- Content provided by‌ FirstRanker.com ---

iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m

--- Content provided by⁠ FirstRanker.com ---

v=
{lS]=1
3
0.51
z

--- Content provided by FirstRanker.com ---

o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

--- Content provided by⁠ FirstRanker.com ---

Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof

--- Content provided by‌ FirstRanker.com ---

p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

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v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3
.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.

--- Content provided by FirstRanker.com ---

.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.
.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.

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.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.

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2L
22.
23.
24.
25.

--- Content provided by‌ FirstRanker.com ---

.Letu=l?

FindvinR?suchthatl
l?3~l
=T

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2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636

--- Content provided by⁠ FirstRanker.com ---

A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof

--- Content provided by‍ FirstRanker.com ---

if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-

--- Content provided by FirstRanker.com ---

$3111i
._.-QIOQNI
b?
Unit-lll
5712

--- Content provided by FirstRanker.com ---

2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[

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Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat

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_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-

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a.Findtheiniageofp(t)=3-2t+l?.

26.
27.
28.

--- Content provided by‍ FirstRanker.com ---

29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?

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islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)

--- Content provided by FirstRanker.com ---

a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=

--- Content provided by‌ FirstRanker.com ---

l_:1andv;=li

Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

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Constructthegeneralsolutionofx?=Ax.A=i

_?

1g1

--- Content provided by‌ FirstRanker.com ---

ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

2.13.1PracticalsQuestionBank
r~
ll.

--- Content provided by FirstRanker.com ---

SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.

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.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.

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.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and

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cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems

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:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.

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Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

l4.
l6.

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l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l

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a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough

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thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.

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Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)

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i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a

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'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=

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Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.

FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank

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LinearAlgebra
Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=

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l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

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v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

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m
v=
{lS]=1
3
0.51

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z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2

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?--2andv;=l7

Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?

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.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

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v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.
...21?2?3

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.FindthediineiisonofthesubspaceHofIRspannedby_5,w.l5.
.LetHbeanll-(llIllCllSlOllIllsubspaceo!?anll-tlllll0ll?0ll?lvectorqmceV.ShowthatH=V.
.ExplainwhythespacePol?allpolynomialsismiin?nitediiiietisioiialspace.
Unit-II
.Ifa-lx7iiiatrixAhasrank3.?iuldiiii.\'ulA.dimRowA.iiiidniiikAT.

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.lla7x5niatrixAhasrank2.?nddiiii.\'iilA.diinRowA.niidmiikAT.
.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

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20.
2L
22.
23.
24.

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25.
.Letu=l?

FindvinR?suchthatl
l?3~l

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=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA

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26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.

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.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl

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iii-FH-
$3111i
._.-QIOQNI
b?
Unit-lll

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5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.
T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.

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LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.
LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear

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transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial

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(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

26.
27.

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28.
29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.

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AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J

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9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l

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LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li

Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

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Constructthegeneralsolutionofx?=Ax.A=i

_?

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1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

2.13.1PracticalsQuestionBank
r~

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ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).
.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).

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.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).
.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)

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andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.

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.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres
;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.

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.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass

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throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

l4.

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l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis

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r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).

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Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation

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312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline

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(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'

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2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis

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12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.

21.

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22.
23.
24.
26.
27.

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Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid

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12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.

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Obtainthetangentplanestotheellipsoid
1
u
Iu||
i-

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+
ada
?EH.
+
?u

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whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface

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i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice

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whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis

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FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I

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I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=

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.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=
iii

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Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m
v=

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{lS]=1
3
0.51
z
o}

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a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

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Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.

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Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find

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ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.
2L

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22.
23.
24.
25.
.Letu=l?

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FindvinR?suchthatl
l?3~l
=T
2438]uv.

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.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6

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-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0

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Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i

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._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l

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T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

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26.
27.
28.
29.

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--- Content provided by‌ FirstRanker.com ---

P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.

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b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li

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Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

Constructthegeneralsolutionofx?=Ax.A=i

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_?

1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

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2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry

--- Content provided by‍ FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

--- Content provided by⁠ FirstRanker.com ---

1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres

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whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H

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Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

l4.
l6.
l8.

--- Content provided by FirstRanker.com ---

20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'

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.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations

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ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto

--- Content provided by‌ FirstRanker.com ---

andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.

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2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'

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showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree

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points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.

21.
22.
23.

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24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline

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l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto

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r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1

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u
Iu||
i-
+
ada

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?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.

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Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie

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~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid

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1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis

28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni

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totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
FirstRanker.com-FirstRanker'sChoice

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2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I
I
'2?l

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l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

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v7=
iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

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m
v=
{lS]=1
3

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0.51
z
o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch

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l?2
?--2andv;=l7

Determineif{vbv?isabasicforR?.ls{vbv?a
3?9

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basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.

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l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find
.l...twodifferentwaystoexpressiasalinearcombinationofv?.v;.v;.
.Findthe(liineiisioiioftliesubspaceofallwctorsin1R3whose?rstandthirdentriesareequal.

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20.
2L
22.
23.

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24.
25.
.Letu=l?

FindvinR?suchthatl

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l?3~l
=T
2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.

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.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6
-l9?l293l2'
i-23633?(ii

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.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0
Azlo054]
000i

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Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i
._.-QIOQNI
b?

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Unit-lll
5712
2ain=i23l
LetB={bhbmba}andD={dhdg}bebasesforvectorspacesVandW.respectively.
LetT:V->ll"benlineartransformationwiththepropertythatT(h|)=3di?5th.

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T(b;)=-d|+6d-,-.T(b;)=Jdq.FindthematrixforTrelativetoBandD.
LetA=PDP"andcomputeA?.whereI?=[
Let?D={dbd?andB={bpbg}bebasesforvectorqiiicesVandll".respectively.Let
T:V?>WbealineartransfonnationwiththepropertythatT(d|)=3b;?13b1,T(d;)=
-2b|+5b,.FindthematrixforTrelativetoBand?D.

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LetB={b|.bg.b;)beabassforavectorspaceVandletT:V?>R7bealinear
transformationwiththepropertythat
_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.

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LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

26.

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27.
28.
29.
30.
b.ShowthatTisalineartransformation.

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c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?
islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)

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PU)J
9(0)
a.ShowthatTisalineartransformation.
b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.

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De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li

Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue

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problemx'=Ax.x(0)=[g

Constructthegeneralsolutionofx?=Ax.A=i

_?

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1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

2.13.1PracticalsQuestionBank

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r~
ll.
SolidGeometry
Unit-l
.FindthecquationofthespherethroughthefourpointsH.?l.2).(0.?2.3).(1.?5.?1).(2.0.l).

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.Findtheequationofthespherethroughthe[ourpoints(0.0.0).(?a.b.c).(a.?b.c).(a.b.?r).
.Findthecentreandtheradiusofthecircle1+2y+2=15.124-yz+22-2y?4:=ll.
.Showthatthefollowingpointsaremncyclie:
(a)(s.o.2).(2.-c.o).(n-rus).(4~_9.6).
(u)(-s.s.2).(-s.2.2)(-'r.o.o).(-4.a.s).

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.Findthecentresofthetwoqihenswhichtouchtheplane41+3y+=47ntthepoints(8,5.4)
andwhichto\|chthespherer?+y?+z?=1.
.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o

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touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres
.r?+y?+:'+.r-3:-2=0.2(:?+y'+:?)+:+3y+4=0.
.Findthelimitingpointsoftheco-axalsystemofspheres

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;?+y?+=?-2o?+aoy-4o=+29+,\(21-sy+4=)=0.
.Findthetquationtothetwospheresoftheco-axalsystems
:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.

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.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

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l4.
l6.
l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0

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bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=

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withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators

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satisfyingtherelation
312?4m:+5n:=0.
Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.

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.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is

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1'?l_y_z-3'
2'a
'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.

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.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=
Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.

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21.
22.
23.
24.
26.

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27.
Unit-lll
Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'

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withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.

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71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1
u
Iu||

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i-
+
ada
?EH.
+

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?u
whichareparalleltothepliuie
lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.

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.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.

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Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.
llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I

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whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis

28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni
totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.

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1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
2.14.1PracticalsQuestionBank
IntegralCalculus
Unit-I

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l.LetR=[-3.3]x[-2.2].Withoutexplicitlyevaluatinganyiteratedintegrals.detenninethe
valueof
f/(1%.mm
n
2.Integratethefunctionf(x.y)=31yovertheregionboundedbyy=3213midy=

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l0.
l3.
.f(:.y.:)=2:?y+z:ll"istheregionhoundedbythecylinder:=
.Integratethefunctionf(x.y)=1+yovertheregionhoundedbyz+y=2andy?-2y-.'r=0.

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.Evaluatefforyrlzl.whereDistheregionboundedby.r=y?andy=1".
.Evaluateffo?zd/i.whereDisthetriangularregionwithvertices(0.0).(l.0)and(1.1).
.Evaluateffbilyrl/l.whereDistheregionboundedbyry?=1.y=r.r=0andy=3.
.EvaluateffD(.-r?2y)d?l.whereDistheregionboundedb_vy=:2+2andy=212?2.
.EvaluateffD(r"?+y')d/l.whereDistheregioninthe?rstquadrantboundedbyy=4r.y=3:

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and.'ry=3.
.Considertheintegral
22:
/(2:+l)rlyd.r
oa

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a)Evaluatethisintegral.
b)Sketchtheregionofintegratiou.
c)Writeanequivalentiteratedintegralwiththeorderofintegrationreverse.Evaluatethis
newintegralandcheek_vouransweragreeswithpart(a).
Findthevolumeoftheregionunderthegraphof

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!(1-u)=2?|1|?|y|
andabovethery-plmre
Unit-II
IntegratethefollowingovertheindicatrxlregionW.
y?.thery-plane.the

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planes1=0.:=l.l!?/&.y=2.
.j(z.y.:)=y:Wistheregionboundedbytheplane1+y+z=2.thecylinder12+:2=l
andy=0.
[(r.y.:)=8ryzzWistheregionhoundedbythecylindery=r2.theplaney+:=9and
thezy-plane.

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FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra
Unit-I

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I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.
.Letv;=

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.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=
iii

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Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m
v=

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{lS]=1
3
0.51
z
o}

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a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

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Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof
p(l)=l+~lf+7|?relativetoB.

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Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

v;=[__3I1spmtlllzbutdonotfomiabasis.Find

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ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.
2L

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22.
23.
24.
25.
.Letu=l?

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FindvinR?suchthatl
l?3~l
=T
2438]uv.

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.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636
A=?2?36-30?6

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-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof
if;?Z?0

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Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-
$3111i

--- Content provided by FirstRanker.com ---

._.-QIOQNI
b?
Unit-lll
5712
2ain=i23l

--- Content provided by‌ FirstRanker.com ---

--- Content provided by FirstRanker.com ---

T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-
a.Findtheiniageofp(t)=3-2t+l?.

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26.
27.
28.
29.

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--- Content provided by FirstRanker.com ---

P(-1)
9(3)
PU)J
9(0)
a.ShowthatTisalineartransformation.

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b.FindthematrixforTrelativetothebases(l,t.t?.t?)forP;andthestandardbasisfor
R4.
De?iieT2P;?>R?byT(p)=l
LetAbea2x2matrixwitheigenvalues-3and-landcorrespondingeigenvcctorsvi=
l_:1andv;=li

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Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

Constructthegeneralsolutionofx?=Ax.A=i

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_?

1g1
ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

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2.13.1PracticalsQuestionBank
r~
ll.
SolidGeometry

--- Content provided by FirstRanker.com ---

--- Content provided by‌ FirstRanker.com ---

1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and
cutsorthogonallythetwospheres

--- Content provided by FirstRanker.com ---

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whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.
Unit-H

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Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

l4.
l6.
l8.

--- Content provided by‌ FirstRanker.com ---

20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l
a:b:'

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.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough
thecurvegivenbytheequations

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andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)
i=-2=.

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2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a
'1'

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--- Content provided by FirstRanker.com ---

points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.

21.
22.
23.

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24.
26.
27.
Unit-lll
Findthepointsofintermctionoftheline

--- Content provided by‌ FirstRanker.com ---

l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.
Findtheequationstothetangentplanesto

--- Content provided by‍ FirstRanker.com ---

r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid
1

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u
Iu||
i-
+
ada

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?EH.
+
?u
whichareparalleltothepliuie
lr+my+n:=0.

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Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.
paralleltothepliuie

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1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis

28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni

--- Content provided by‍ FirstRanker.com ---

totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.
2.14.1PracticalsQuestionBank

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IntegralCalculus
Unit-I
l.LetR=[-3.3]x[-2.2].Withoutexplicitlyevaluatinganyiteratedintegrals.detenninethe
valueof
f/(1%.mm

--- Content provided by‍ FirstRanker.com ---

n
2.Integratethefunctionf(x.y)=31yovertheregionboundedbyy=3213midy=

l0.
l3.

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.f(:.y.:)=2:?y+z:ll"istheregionhoundedbythecylinder:=
.Integratethefunctionf(x.y)=1+yovertheregionhoundedbyz+y=2andy?-2y-.'r=0.
.Evaluatefforyrlzl.whereDistheregionboundedby.r=y?andy=1".
.Evaluateffo?zd/i.whereDisthetriangularregionwithvertices(0.0).(l.0)and(1.1).
.Evaluateffbilyrl/l.whereDistheregionboundedbyry?=1.y=r.r=0andy=3.

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.EvaluateffD(.-r?2y)d?l.whereDistheregionboundedb_vy=:2+2andy=212?2.
.EvaluateffD(r"?+y')d/l.whereDistheregioninthe?rstquadrantboundedbyy=4r.y=3:
and.'ry=3.
.Considertheintegral
22:

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/(2:+l)rlyd.r
oa
a)Evaluatethisintegral.
b)Sketchtheregionofintegratiou.
c)Writeanequivalentiteratedintegralwiththeorderofintegrationreverse.Evaluatethis

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newintegralandcheek_vouransweragreeswithpart(a).
Findthevolumeoftheregionunderthegraphof
!(1-u)=2?|1|?|y|
andabovethery-plmre
Unit-II

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IntegratethefollowingovertheindicatrxlregionW.
y?.thery-plane.the
planes1=0.:=l.l!?/&.y=2.
.j(z.y.:)=y:Wistheregionboundedbytheplane1+y+z=2.thecylinder12+:2=l
andy=0.

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[(r.y.:)=8ryzzWistheregionhoundedbythecylindery=r2.theplaney+:=9and
thezy-plane.

l7.
l9.

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20.
2
r~
22.
23.

--- Content provided by⁠ FirstRanker.com ---

24.
25.
.f(.'r.y.:)=z:Wistheregioninthe?rstoctantboundvxlbythecylindery:+z?=9andthe
planesy=1.:=0andz=0.
.f(r.y.z)=l?s?:Wisthetetrahedronwithvertiees(0.0.0).(1.0.0).(0.2.0)and(0.0.3).

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.j(r.y.:)=3r;II"istheregioninthe?rstoctantboundedby:.=r2+5/2.:=0.y=0and
z=-l.
j(x.y.z)=r+y:Wistheregionboundedbythecylinderr2+3:2=9andtheplaney=0.
r+y=3.
.[(1.y.z)=z:II?istheregionboundedbyz=0.I?:+4y?=-lturdz=1+2.

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Unit-III
IntegratethefollowingovertheindicatedregionW.
](.-r.y.:)=-l.r+yzWistheregionboundedby1'=y?.y=:..r=yaadz=0.
j(z.y.z)=.r:Wistheregioninthe?rstoctantboundedbyz=r2+2f.z=G
?

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:2
?
y?.
r=0arrdy=0.
LetT(u.r')=(3ll.?v)

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.WriteT(u.u)asAIt;lforasuitablematrixA.
DeztribetheimageD=T(D?).whereD?istheunitsquare[0.1]x[0.1].
f/?li?
1?2y
D

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whereDistheregioninR2mrelosetlbytheIirresy=

y=0and
I+
y=I.

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Deterrrrinethevalueof
Evaluate
(2.r+y?3)?
tld.
(2y?:r+6)?Iy

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whereDisthesquarewithwrtices(0.0).(2.l).(3.-l)iurd(l.-2).(Hint:FirstdtetchDand
?ndtheequationsofitssides)
Evaluate
f]cw?rz+yzM/l.
0

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whereDistheshadedregioninthefollowing?gure-l.

FirstRanker.com-FirstRanker'sChoice
2.12.1PracticalsQuestionBank
LinearAlgebra

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Unit-I
I
'2?l
l.LetHbethesetofallvectorsoftheforniFindavectorvin1R3suchthatH=
l4.

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.Letv;=
.ThesetB={l?l7.!?12.2?1+l2}isabasisforP2.
.Thevmtorv;=I_:li

v7=

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iii
Span{v).\\'h_vdoesthis?IOW'thatHisasubspaceoflRJ?
.i.?vi?lhCas?quadrantinthery-pliuie:that

m

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v=
{lS]=1
3
0.51
z

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o}
a.IfuandvareinV.isu+vinV?Why?
b.Findaspeci?cvectoruinViuidaspeci?cscalarcsuch
l?2
?--2andv;=l7

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Determineif{vbv?isabasicforR?.ls{vbv?a
3?9
basislorlll2'.?
.ThesctB={l+12.!+12.1+2t+12}isabasisForP1.Fiiidthecoordinatevectorof

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p(l)=l+~lf+7|?relativetoB.
Findthecoordinatewctorof
p(|)=1+31-6i?relativetoB.
l_?

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.lftheiiullspaeeofan8x5inatrixAisZl-diiueiiiional.whatisthedimensionoftherowspace
ofA?
IfAisa3x7matrix.whatisthesniallmtpossiblediniensioiiof.\'iilA?

20.

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2L
22.
23.
24.
25.

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.Letu=l?

FindvinR?suchthatl
l?3~l
=T

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2438]uv.
.lfAisa7X5matrix.whatisthelargestpossiblerankofA?lfAisa5x7matrix.whatis
thelargestpossiblerankofA?Explain_vouranswers.
.Withoutcalculations.listrankAanddiin.\'ulA
26?6636

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A=?2?36-30?6
-l9?l293l2'
i-23633?(ii
.UseapropertyofdcteriiiinantstoshowthatAandAThavethesamecharacteristicpolynomial.
.Findthecharacteristicequationof

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if;?Z?0
Azlo054]
000i
Findthecharacteristicpolynomialandtherealeigenvaluesofl
iii-FH-

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$3111i
._.-QIOQNI
b?
Unit-lll
5712

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_2:1?3r;+1,
T(.r|b|+13b;+.'r,b;)-_2rl+51.?
FindthematrixforTrelativetoBandthestandardbasisforR2.
LetT:P;?>P;bethetmnsformationthatmapsapolynomialp(t)intothepolynomial
(f+3)P(')-

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a.Findtheiniageofp(t)=3-2t+l?.

26.
27.
28.

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29.
30.
b.ShowthatTisalineartransformation.
c.FindthematrixforTrelativetothebases{LL17}and{LLFJJ}.
AssumethemappingT:1?;?>ll?;de?nedbyT(?Q+?|l+l|;lz)=3ao+(5no?2zii)l+(~lrii+01)!?

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islinear.FindthematrixrepresentationofTrdativetothebasisB=(l.Li?).
P(-1)
9(3)
PU)J
9(0)

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l_:1andv;=li

Letx(t)bethepositionofaparticleattimet.Solvetheinitialvalue
problemx'=Ax.x(0)=[g

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Constructthegeneralsolutionofx?=Ax.A=i

_?

1g1

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ComputetheorthogonalprojectionoflQ1ontothelinethroughl_;1andtheorigin.

2.13.1PracticalsQuestionBank
r~
ll.

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.Showthatthespheres
1?+y?+=?=25
1?+y?+=?-24:-40y-1s=+22s=o
touchexternallyand?ndthepointofthecontact.
.Findtheequationofthespherethatpassesthroughthetwopoints(0,3.0)_(?2,?l,?1l)and

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:'+y?+;'-s+,\(2:+y+a=-s)=0.
whichtouchtheplane
3r+-ly=l5.
.Showthattheradicalplanesofthesphereofaco-axalsystemandofanygivenspherepass
throughaline.

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Unit-H
Findtheequationoftheconewhosevertmcisthepoint(l.I.0)andwhomguidingcurveis

l4.
l6.

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l8.
20.
.Thesectionofaconewhosevertu:isPandguidingcurvetheellipse13/02+yz/b:=L:=0
bytheplane.r=0isarectangularhyperbola.ShowthatthelocusofPis
r_2+y2+:7=l

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a:b:'
.Findtheenvelopingconeofthesphere
.r?+y?+z?-2:+4:=
withitsvatmcat(l.l.1).
Findtheequationofthequadricconewhosevertexisattheoriginandwhichpassesthrough

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thecurvegivenbytheequations
ax?+6512+0.22=1.1:+my+nz=
.Findtheequationoftheconewithvertexattheoriginanddirectioncosinesofitsgenerators
satisfyingtherelation
312?4m:+5n:=0.

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Findtheequationofthecylinderwhosegeneratorsareparallelto
andwhoseguidingcurveistheellipse
12+2y2=l.:=3.
.Findtheequationoftherightcircularcylinderofmdius2whoseaxisistheline
(I-l)(=-3)

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i=-2=.
2(u)2
Theaxisofarightcircularcylinderofradius2is
1'?l_y_z-3'
2'a

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'1'
showthatitsequationis
|0r?+5112+13:?-l2ry?(iyz--lz.r-8.'r+30y-"u:+ss=0.
.Findtheequationofthecircularcylinderwhoseguidingcircleis
12+yz+z?9=0.r?y+z=

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Obtaintheequationoftheriglicircularcylinderdescribedonthecirclethroughthethree
points(l.0.0).(0.1.0).(0.0.l)asguidingcircle.

21.
22.

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23.
24.
26.
27.
Unit-lll

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Findthepointsofintermctionoftheline
l<:+5)?
s?i?1'
withthecoiiieoid
12a?-l7y2+n?=1.

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Findtheequationstothetangentplanesto
r?-a?-=?+21=0.
whichpassthroughtheline.
71-Gy+9=3.:=3.
Obtainthetangentplanestotheellipsoid

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1
u
Iu||
i-
+

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ada
?EH.
+
?u
whichareparalleltothepliuie

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lr+my+n:=0.
Showthattheplane3r+l2y?G:?l7=0touchestheconicoid3.1-2?6y?+9:2+l7=U.
and?ndthepointofcontact.
.Findtheequationstotlietangentpliuiestothesurface
i1?may?+1=?+l3=o.

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paralleltothepliuie
~lr+20y?2l:=0.
Findtheirpointsofcontactalso.
Findthelocusoftheperpendicularsfromtheorigintothetangentplanestothesnr?ice
whichcutoflfromitsaxesinterceptsthesumofwhosereciprocalsisequaltoaconstantl/k.

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llthesectionoftheenvelopingconeoftheellipsoid
1.2y::2l
n2+b2+c7_I
whosevertexisPbytheplanez=0isarectangularhyperbola.showthatthelocusofPis

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28.Findthelocusofpointsfromwhichthreemutuallyperpendiculartangentlinescanhedravni
totheconieoidor?+by?+cz?=l.
29.P(l.3.2)isapointontheconicoid.
1?-2y?+s=?+s=o.
Findthelocusoi?themid-pointsofchordsdrawnparalleltoOP.

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2.14.1PracticalsQuestionBank
IntegralCalculus
Unit-I
l.LetR=[-3.3]x[-2.2].Withoutexplicitlyevaluatinganyiteratedintegrals.detenninethe
valueof

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f/(1%.mm
n
2.Integratethefunctionf(x.y)=31yovertheregionboundedbyy=3213midy=

l0.

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l3.
.f(:.y.:)=2:?y+z:ll"istheregionhoundedbythecylinder:=
.Integratethefunctionf(x.y)=1+yovertheregionhoundedbyz+y=2andy?-2y-.'r=0.
.Evaluatefforyrlzl.whereDistheregionboundedby.r=y?andy=1".
.Evaluateffo?zd/i.whereDisthetriangularregionwithvertices(0.0).(l.0)and(1.1).

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.Evaluateffbilyrl/l.whereDistheregionboundedbyry?=1.y=r.r=0andy=3.
.EvaluateffD(.-r?2y)d?l.whereDistheregionboundedb_vy=:2+2andy=212?2.
.EvaluateffD(r"?+y')d/l.whereDistheregioninthe?rstquadrantboundedbyy=4r.y=3:
and.'ry=3.
.Considertheintegral

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22:
/(2:+l)rlyd.r
oa
a)Evaluatethisintegral.
b)Sketchtheregionofintegratiou.

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c)Writeanequivalentiteratedintegralwiththeorderofintegrationreverse.Evaluatethis
newintegralandcheek_vouransweragreeswithpart(a).
Findthevolumeoftheregionunderthegraphof
!(1-u)=2?|1|?|y|
andabovethery-plmre

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Unit-II
IntegratethefollowingovertheindicatrxlregionW.
y?.thery-plane.the
planes1=0.:=l.l!?/&.y=2.
.j(z.y.:)=y:Wistheregionboundedbytheplane1+y+z=2.thecylinder12+:2=l

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andy=0.
[(r.y.:)=8ryzzWistheregionhoundedbythecylindery=r2.theplaney+:=9and
thezy-plane.

l7.

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l9.
20.
2
r~
22.

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23.
24.
25.
.f(.'r.y.:)=z:Wistheregioninthe?rstoctantboundvxlbythecylindery:+z?=9andthe
planesy=1.:=0andz=0.

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.f(r.y.z)=l?s?:Wisthetetrahedronwithvertiees(0.0.0).(1.0.0).(0.2.0)and(0.0.3).
.j(r.y.:)=3r;II"istheregioninthe?rstoctantboundedby:.=r2+5/2.:=0.y=0and
z=-l.
j(x.y.z)=r+y:Wistheregionboundedbythecylinderr2+3:2=9andtheplaney=0.
r+y=3.

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.[(1.y.z)=z:II?istheregionboundedbyz=0.I?:+4y?=-lturdz=1+2.
Unit-III
IntegratethefollowingovertheindicatedregionW.
](.-r.y.:)=-l.r+yzWistheregionboundedby1'=y?.y=:..r=yaadz=0.
j(z.y.z)=.r:Wistheregioninthe?rstoctantboundedbyz=r2+2f.z=G

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?
:2
?
y?.
r=0arrdy=0.

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LetT(u.r')=(3ll.?v)
.WriteT(u.u)asAIt;lforasuitablematrixA.
DeztribetheimageD=T(D?).whereD?istheunitsquare[0.1]x[0.1].
f/?li?
1?2y

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D
whereDistheregioninR2mrelosetlbytheIirresy=

y=0and
I+

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y=I.
Deterrrrinethevalueof
Evaluate
(2.r+y?3)?
tld.

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(2y?:r+6)?Iy
whereDisthesquarewithwrtices(0.0).(2.l).(3.-l)iurd(l.-2).(Hint:FirstdtetchDand
?ndtheequationsofitssides)
Evaluate
f]cw?rz+yzM/l.

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0
whereDistheshadedregioninthefollowing?gure-l.

Arr:of|rcircle
ofradiusl.i,

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(ceiilenaat'
"Iilli?l
\

Figurel:

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2G.Evaluatel
[Itl?.
0
WhereDisthediskofradiuslwithcerterat(0.l).(BecarefulwhenyoudeatribeD.)
27.Determinethevalueof_

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iv
whereWisthesolidregionboundedbytheplanez=12andtheparaboloidz=2r?+2y?
?
6.
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This post was last modified on 21 November 2019

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